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Integrability properties and Limit Theorems for the exit time from a cone of planar Brownian motion
Stavros Vakeroudis, Marc Yor
To cite this version:
Stavros Vakeroudis, Marc Yor. Integrability properties and Limit Theorems for the exit time from a cone of planar Brownian motion. 2011. �hal-00654695�
Integrability properties and Limit Theorems for the exit time from a cone of planar Brownian motion
S. Vakeroudis ∗† M. Yor ∗‡
December 22, 2011
Abstract
We obtain some integrability properties and some limit Theorems for the exit time from a cone of a planar Brownian motion, and we check that our computations are correct via Bougerol’s identity.
Key words: Bougerol’s identity, planar Brownian motion, skew-product representation, exit time from a cone.
MSC Classification (2010): 60J65, 60F05.
1 Introduction
We consider a standard planar Brownian motion§ (Zt = Xt+iYt, t ≥ 0), starting from x0 +i0, x0 > 0, where (Xt, t ≥ 0) and (Yt, t ≥ 0) are two independent linear Brownian motions, starting respectively from x0 and 0.
As is well known [ItMK65], since x0 6= 0, (Zt, t ≥ 0) does not visit a.s. the point 0 but keeps winding around 0 infinitely often. In particular, the continuous winding process θt = Im(Rt
0 dZs
Zs ), t ≥ 0 is well defined. A scaling argument shows that we may assume x0 = 1, without loss of generality, since, with obvious notation:
Zt(x0), t≥0(law)
=
x0Z(t/x(1) 2
0), t≥0
. (1)
∗Laboratoire de Probabilités et Modèles Aléatoires (LPMA) CNRS : UMR7599, Université Pierre et Marie Curie - Paris VI, Université Paris-Diderot - Paris VII, 4 Place Jussieu, 75252 Paris Cedex 05, France. E-mail: stavros.vakeroudis@etu.upmc.fr
†Probability and Statistics Group, School of Mathematics, University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 9PL, United Kingdom.
‡Institut Universitaire de France, Paris, France. E-mail: yormarc@aol.com
§When we simply write: Brownian motion, we always mean real-valued Brownian motion, starting from 0. For 2-dimensional Brownian motion, we indicate planar or complex BM.
Thus, from now on, we shall take x0 = 1.
Furthermore, there is the skew product representation:
log|Zt|+iθt ≡ Z t
0
dZs Zs
= (βu+iγu)
u=Ht=Rt 0
ds
|Zs|2
, (2)
where (βu+iγu, u≥0) is another planar Brownian motion starting from log 1 +i0 = 0.
Thus, the Bessel clock H plays a key role in many aspects of the study of the winding number process (θt, t ≥0) (see e.g. [Yor80]).
Rewriting (2) as:
log|Zt|=βHt; θt =γHt, (3) we easily obtain that the two σ-fields σ{|Zt|, t ≥ 0} and σ{βu, u ≥ 0} are identical, whereas (γu, u≥0) is independent from(|Zt|, t ≥0).
We shall also use Bougerol’s celebrated identity in law [Bou83, ADY97] and [Yor01] (p.
200), which may be written as:
for fixed t, sinh(βt)(law)= ˆβAt(β) (4) where (βu, u ≥ 0) is 1-dimensional BM, Au(β) = Ru
0 dsexp(2βs) and ( ˆβv, v ≥ 0) is another BM, independent of (βu, u ≥ 0). For the random times Tc|θ| ≡ inf{t : |θt| = c}, and Tc|γ| ≡ inf{t : |γt| = c}, (c > 0) by using the skew-product representation (3) of planar Brownian motion [ReY99], we obtain:
Tc|θ| =AT|γ|
c (β)≡ Z Tc|γ|
0
dsexp(2βs) = Hu−1 u=Tc|γ|
. (5)
Moreover, it has been recently shown that, Bougerol’s identity applied with the random time Tc|θ| instead oft in (4) yields the following [Vak11]:
Proposition 1.1 The distribution of Tc|θ| is characterized by its Gauss-Laplace trans- form:
E
"s 2c2 πTc|θ|
exp
− x 2Tc|θ|
#
= 1
√1 +xϕm(x), (6) for every x≥0, with m= 2cπ, and:
ϕm(x) = 2
(G+(x))m+ (G−(x))m, G±(x) =√
1 +x±√
x. (7)
The remainder of this article is organized as follows: in Section 2 we study some inte- grability properties for the exit times from a cone; more precisely we obtain some new results concerning the negative moments of Tc|θ| and of Tcθ ≡inf{t:θt=c}. In Section 3 we state and prove some limit Theorems for these random times forc→0and for c→ ∞ followed by several generalizations (for extensions of these works to more general planar processes, see e.g. [DoV12]). We use these results in order to obtain (see Remark 3.4) a
new simple non-computational proof of Spitzer’s celebrated asymptotic Theorem [Spi58], which states that:
2 logt θt
(law)
t→∞−→C1 , (8)
with C1 denoting a standard Cauchy variable (for other proofs, see also e.g. [Wil74, Dur82, MeY82, BeW94, Yor97, Vak11]). Finally, in Section 4 we use the Gauss-Laplace transform (6) which is equivalent to Bougerol’s identity (4) in order to check our results.
2 Integrability Properties
Concerning the moments of Tc|θ|, we have the following (a more extended discussion is found in e.g. [MaY05]):
Theorem 2.1 For every c >0, Tc|θ| enjoys the following integrability properties:
(i) for p >0, Eh Tc|θ|
pi
<∞, if and only if p < 4cπ. (ii) for any p <0, Eh
Tc|θ|
pi
<∞.
Corollary 2.2 For 0< c < d, the random times T−d,cθ ≡ inf{t : θt ∈/ (−d, c)}, Tc|θ| and Tcθ satisfy the inequality:
Tcθ ≥T−d,cθ ≥Tc|θ|. (9)
Thus, their negative moments satisfy:
f or p >0, E 1
(Tcθ)p
≤E
"
1 T−d,cθ p
#
≤E
1
Tc|θ|
p
<∞. (10) Proofs of Theorem 2.1 and of Corollary 2.2
(i) The original proof is given by Spitzer [Spi58], followed later by many authors [Wil74, Bur77, MeY82, Dur82, Yor85]. See also [ReY99] Ex. 2.21/page 196.
(ii) In order to obtain this result, we might use the representation Tc|θ| =AT|γ|
c together with a recurrence formula for the negative moments of At [Duf00], Theorem 4.2, p. 417 (in fact, Dufresne also considers A(µ)t = Rt
0 dsexp(2βs+ 2µs), but we only need to take µ= 0 for our purpose, and we note At ≡ A(0)t ) [Vakth11]. However, we can also obtain this result by simply remarking that the RHS of the Gauss-Laplace transform (6) in Proposition 1.1 is an infinitely differentiable function in 0 (see also [VaY11]), thus:
E
1 Tc|θ|
p
<∞, for every p >0. (11) Now, Corollary 2.2 follows immediately from Theorem 2.1 (ii).
3 Limit Theorems for T
c|θ|3.1 Limit Theorems for Tc|θ|, as c → 0 and c → ∞
The skew-product representation of planar Brownian motion allows to prove the three following asymptotic results for Tc|θ|:
Proposition 3.1 a) Forc→0, we have:
1
c2 Tc|θ|(law)−→c→0 T1|γ|. (12)
b) For c→ ∞, we have:
1
c log Tc|θ| (law)
c→∞−→2|β|T1|γ|. (13)
c) Forε→0, we have:
1 ε2
Tc+ε|θ| −Tc|θ|(law)
−→ε→0 exp 2βT|γ|
c
T1γ′, (14) whereγ′ stands for a real Brownian motion, independent fromγ, andT1γ′ = inf{t:γt′ = 1} Proof of Proposition 3.1:
We rely upon (5) for the three proofs. By using the scaling property of BM, we obtain:
Tc|θ|=AT|γ|
c (β)(law)= Au(β)
u=c2T1|γ|
thus:
1
c2Tc|θ|(law)= Z T1|γ|
0
dv exp (2cβv) . (15)
a)Forc→0, the RHS of (15) converges toT1|γ|, thus we obtain part a)of the Proposition.
b) Forc→ ∞, taking logarithms on both sides of (15) and dividing by c, on the LHS we obtain 1c log
Tc|θ|
− 2c logcand on the RHS:
1 c log
Z T1|γ|
0
dv exp (2cβv)
!
= log
Z T1|γ|
0
dv exp (2cβv)
!1/c
,
which, from the classical Laplace argument: kfkpp→∞−→kfk∞, converges for c → ∞, to- wards:
2 sup
v≤T1|γ|
(βv)(law)= 2|β|T1|γ|.
This proves part b) of the Proposition.
c)
Tc+ε|θ| −Tc|θ| =
Z Tc+ε|γ|
Tc|γ|
du exp (2βu) =
Z Tc+ε|γ|−Tc|γ|
0
dv exp 2βT|γ|
c
exp 2
βv+T|γ|
c −βT|γ|
c
= exp 2βT|γ|
c
Z Tc+ε|γ|−Tc|γ|
0
dv exp (2Bv), (16)
where
Bs≡βs+T|γ|
c −βT|γ|
c , s≥0
is a BM independent of Tc|γ|.
We study nowT˜c,c+ε|γ| ≡Tc+ε|γ| −Tc|γ|, the first hitting time of the levelc+εfrom|γ|, starting from c. Thus, we define: ρu ≡ |γu|, starting also from c. Thus, ρu =c+δu+Lu, where (δs, s≥0) is a BM and (Ls, s≥0)is the local time of ρ at 0. Thus:
T˜c,c+ε|γ| = inf{u≥0 :ρu =c+ε} ≡inf{u≥0 :δu+Lu =ε}
u=ε2v
= ε2inf
v ≥0 : 1
εδvε2 + 1
εLvε2 = 1
. (17)
From Skorokhod’s Lemma [ReY99]:
Lu = sup
y≤u
((−c−δy)∨0) we deduce:
1
εLvε2 = sup
y≤vε2
((−c−δy)∨0)y=ε
2σ
= sup
σ≤v
−c−ε1 εδσε2
∨0
= 0. (18)
Hence, with γ′ denoting a new BM independent from γ, (16) writes:
Tc+ε|θ| −Tc|θ| = exp 2βT|γ|
c
Z ε2T1γ′ 0
dv exp (2Bv). (19) Thus, dividing both sides of (19) by ε2 and making ε → 0, we obtain part c) of the Proposition.
Remark 3.2 The asymptotic result c) in Proposition 3.1 may also be obtained in a straightforward manner from (16) by analytic computations. Indeed, using the Laplace transform of the first hitting time of a fixed level by the absolute value of a linear Brownian motion Eh
e−λ
2 2 Tb|γ|i
= cosh(λb)1 (see e.g. Proposition 3.7, p 71 in Revuz and Yor [ReY99]), we have that for 0< c < b, and λ≥0:
E
e−λ
2 2
Tb|γ|−Tc|γ|
= cosh(λc)
cosh(λb) (20)
Using now b=c+ε, for every ε >0, the latter equals:
cosh(λcε) cosh λε(c+ε)
−→ε→0 e−λ.
The result follows now by remarking that e−λ is the Laplace transform (for the argument λ2/2) of the first hitting time of 1 by a linear Brownian motion γ′, independent from γ.
3.2 Generalizations
Obviously we can obtain several variants of Proposition 3.1, by studying T−bc,acθ , 0 <
a, b≤ ∞, for c→0 orc→ ∞, anda, b fixed. We define T−d,cγ ≡inf{t :γt∈/ (−d, c)} and we have:
• c12 T−bc,acθ (law)−→c→0 T−b,aγ .
• 1c log T−bc,acθ (law)
c→∞−→2|β|T−b,aγ .
In particular, we can take b=∞, hence:
Corollary 3.3 a) For c→0, we have:
1
c2 Tacθ (law)−→c→0 Taγ. (21)
b)For c→ ∞, we have:
1
c log Tacθ(law)
c→∞−→ 2|β|Taγ
(law)
= 2|Ca|, (22)
where (Ca, a≥0) is a standard Cauchy process.
Remark 3.4 (Yet another proof of Spitzer’s Theorem)
Taking a = 1, from Corollary 3.3(b), we can obtain yet another proof of Spitzer’s cele- brated asymptotic Theorem stated in (8). Indeed, (22) can be equivalently stated as:
P logTcθ < cx(law)
c→∞−→ P(2|C1|< x). (23) Now, the LHS of (23) equals:
P logTcθ < cx
≡ P Tcθ <exp(cx)
≡P sup
u≤exp(cx)
θu > c
!
= P |θexp(cx)|> c
=P
|θt|> logt x
, (24)
with t = exp(cx). Thus, because |C1|(law)= |C1|−1, (23) now writes:
f or every x >0 given, P
|θt|> logt x
(law)
−→t→∞ P
|C1|> 2 x
, (25)
which yields precisely Spitzer’s Theorem (8).
3.3 Speed of convergence
We can easily improve upon Proposition 3.1 by studying the speed of convergence of the distribution of c12 Tc|θ| towards that ofT1|γ|, i.e.:
Proposition 3.5 For any function ϕ∈ C2, with compact support, 1
c2
E
ϕ 1
c2Tc|θ|
−Eh ϕ
T1|γ|i
−→c→0E
ϕ′
T1|γ|
T1|γ|2
+2 3ϕ′′
T1|γ|
T1|γ|3 .(26)
Proof of Proposition 3.5:
We develop exp (2cβv), forc→0, up to the second order term, i.e.:
e2cβv = 1 + 2cβv+ 2c2βv2+. . . .
More precisely, we develop up to the second order term, and we obtain:
E
ϕ 1
c2Tc|θ|
= E
"
ϕ
Z T1|γ|
0
dv exp (2cβv)
!#
= E
"
ϕ T1|γ|
+ϕ′
T1|γ| Z T1|γ|
0
2cβv + 2c2βv2 dv
#
+1 2E
ϕ′′
T1|γ|
4c2
Z T1|γ|
0
βvdv
!2
+c2o(c).
We then remark that Eh Rt
0 βvdvi
= 0, Eh Rt
0βv2dvi
= t2/2 and E
Rt
0βvdv2
= t3/3, thus we obtain (26).
4 Checks via Bougerol’s identity
So far, we have not made use of Bougerol’s identity (4), which helps us to characterize the distribution of Tc|θ|[Vak11]. In this Subsection, we verify that writing the Gauss-Laplace transform in (6) as:
E
r2
π 1 q1
c2Tc|θ|
exp
− xc2 2Tc|θ|
= 1
√1 +xc2ϕm(xc2), (27)
with m =π/(2c), we find asymptotically for c→0 the Gauss-Laplace transform ofT1|γ|. Indeed, from (27), for c→0, we obtain:
E
r2
π 1 q
T1|γ|
exp − x 2T1|γ|
!
= lim
c→0
2 √
1 +xc2+√
xc2π/2c
+√
1 +xc2−√
xc2π/2c. (28)
Let us now study:
√
1 +xc2+√
xc2π/2c
= expπ 2clogh
1 +√
1 +xc2−1 +√
xc2i
∼ exp π
2c
c√
x+ xc2 2
−→c→0 exp π√
x 2
.
A similar calculation finally gives:
E
r2
π 1 q
T1|γ|
exp − x 2T1|γ|
!
= 1 cosh π2√
x , (29)
a result which is in agreement with the law of βT|γ|
1 , whose density is:
E
1 q
2πT1|γ|
exp − y2 2T1|γ|
!
= 1
2 cosh π2y . (30)
Indeed, the law of βT|γ|
c may be obtained from its characteristic function which is given by [ReY99], page 73:
Eh
exp(iλβT|γ|
c )i
= 1
cosh(λc) .
It is well known that [Lev80, BiY87]:
Eh
exp(iλβT|γ|
c )i
= 1
cosh(λc) = 1
cosh(πλπc) = Z ∞
−∞
ei(λcπ)y 1 2π
1 cosh(y2) dy
x=cyπ
=
Z ∞
−∞
eiλx 1 2π
π c
cosh(xπ2c) dx= Z ∞
−∞
eiλx 1 2c
1
cosh(xπ2c) dx . (31) So, the density h−c,c of βT|γ|
c is:
h−c,c(y) = 1
2c
1 cosh(yπ2c) =
1 c
1 eyπ2c +e−yπ2c , and for c= 1, we obtain (30).
We recall from Remark 3.2 that (see also [PiY03], where further results concerning the infinitely divisible distributions generated by some Lévy processes associated with the hyperbolic functionscosh, sinh and tanh can also be found):
E
exp
−λ2 2 Tc|γ|
= 1
cosh(λc) , (32)
thus, forc= 1 and λ= π2√
x, (29) now writes:
E
r2
π 1 q
T1|γ|
exp − x 2T1|γ|
!
=E
exp
−xπ2 8 T1|γ|
, (33)
a result which gives a probabilistic proof of the reciprocal relation in [BPY01] (using the notation of this article, Table 1, p.442):
fC1(x) = 2
πx 3/2
fC1
4 π2x
.
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